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The **Addition Property of Equality** states that if the same quantity is added to both sides of an equation, the two sides remain equal. Mathematically, for any numbers $$a$$, $$b$$, and $$c$$:

If $$a = b$$,
then $$a + c = b + c$$

This property is a fundamental tool used to solve equations where a number is subtracted from the variable. To isolate the variable and "undo" the subtraction, the same number is added to both sides. For example, when solving an equation like $$x - 5 = 8$$, adding $$5$$ to both sides isolates $$x$$ while maintaining the equality.

Addition Property of Equality

Solving the equation $n - 0.63 = -4.2$ involves isolating the variable $n$ and working with decimals. First, apply the Addition Property of Equality by adding ${0.63}$ to both sides, which gives $n - 0.63 + 0.63 = -4.2 + 0.63$. This simplifies the left side to $n$. Adding the decimals on the right side yields $n = -3.57$. To check the solution, substitute ${-3.57}$ back into the original equation for $n$, which gives $-3.57 - 0.63 = -4.2$. Subtracting on the left side produces $-4.2 = -4.2$, a true statement confirming that $n = -3.57$ is the correct solution.

Solving $n - 0.63 = -4.2$ Using the Addition Property

To solve the equation $$a - 28 = -37$$, the objective is to isolate the variable $$a$$ by reversing the subtraction of $$28$$. Apply the Addition Property of Equality by adding $$28$$ to both sides:

$$a - 28 + 28 = -37 + 28$$

On the left side, $$-28 + 28 = 0$$, leaving just $$a$$. On the right side, $$-37 + 28 = -9$$. This simplifies to $$a = -9$$.

To verify, substitute $$-9$$ back into the original equation: $$-9 - 28 = -37$$, which simplifies to $$-37 = -37$$. Since this is a true statement, $$a = -9$$ is confirmed as the correct solution.

Solving $$a - 28 = -37$$ Using the Addition Property

To solve $$x - \frac{5}{8} = \frac{3}{4}$$, the goal is to isolate $$x$$ by undoing the subtraction of $$\frac{5}{8}$$. Apply the Addition Property of Equality by adding $$\frac{5}{8}$$ to both sides:

$$x - \frac{5}{8} + \frac{5}{8} = \frac{3}{4} + \frac{5}{8}$$

On the left side, $$-\frac{5}{8} + \frac{5}{8} = 0$$, leaving just $$x$$. On the right side, the fractions $$\frac{3}{4}$$ and $$\frac{5}{8}$$ have different denominators, so find the least common denominator (LCD) of $$4$$ and $$8$$, which is $$8$$. Rewrite $$\frac{3}{4}$$ as the equivalent fraction $$\frac{6}{8}$$:

$$x = \frac{6}{8} + \frac{5}{8} = \frac{11}{8}$$

To verify, substitute $$\frac{11}{8}$$ back into the original equation:

$$\frac{11}{8} - \frac{5}{8} = \frac{6}{8} = \frac{3}{4}$$

Since $$\frac{3}{4} = \frac{3}{4}$$ is a true statement, $$x = \frac{11}{8}$$ is confirmed as the correct solution.

Solving $$x - \frac{5}{8} = \frac{3}{4}$$ Using the Addition Property

To solve $$8n - 4 = -2n + 6$$, notice that variable terms ($$8n$$ and $$-2n$$) and constant terms ($$-4$$ and $$6$$) appear on **both** sides of the equation. Because $$8 > -2$$, designate the left side as the **variable side** and the right side as the **constant side**.

**Step 1 — Remove the variable term from the constant side:** The term $$-2n$$ is on the constant side, so add $$2n$$ to both sides using the Addition Property of Equality:

$$8n + 2n - 4 = -2n + 2n + 6$$

Combine like terms: $$10n - 4 = 6$$. Now the variable appears only on the left.

**Step 2 — Remove the constant from the variable side:** The constant $$-4$$ is on the variable side, so add $$4$$ to both sides:

$$10n - 4 + 4 = 6 + 4$$

$$10n = 10$$

**Step 3 — Isolate the variable using the Division Property of Equality:** Because $$n$$ is multiplied by the coefficient $$10$$, divide both sides by $$10$$:

$$\frac{10n}{10} = \frac{10}{10}$$

$$n = 1$$

**Step 4 — Check by substitution:** Replace $$n$$ with $$1$$ in the original equation:

$$8(1) - 4 \stackrel{?}{=} -2(1) + 6$$

$$8 - 4 \stackrel{?}{=} -2 + 6$$

$$4 = 4 \checkmark$$

Because both sides are equal, $$n = 1$$ is confirmed as the correct solution. Unlike the earlier examples that used the Subtraction Property to collect terms, this equation requires the **Addition Property** in both collecting steps — adding $$2n$$ to eliminate the negative variable term from the right side, and adding $$4$$ to eliminate the negative constant from the left side. Additionally, a final **division** step is needed because the combined coefficient ($$10$$) is not equal to $$1$$.

Solving $$8n - 4 = -2n + 6$$ by Collecting Variables and Constants

The **Extended Addition Property of Equality** generalizes the basic Addition Property by allowing **different but equal** quantities to be added to the two sides of an equation, rather than requiring the same quantity on both sides. Formally, for any expressions $$a$$, $$b$$, $$c$$, and $$d$$:

If $$a = b$$ and $$c = d$$, then $$a + c = b + d$$.

The basic Addition Property is a special case of this rule (where $$c$$ and $$d$$ happen to be the same value). The extended version is essential to the elimination method for solving systems of equations because it justifies adding the left-hand sides and right-hand sides of two different equations together: each equation is a statement of equality, so the two left sides and two right sides are equal quantities, and combining them preserves equality. For example, given $$3x + y = 5$$ and $$2x - y = 0$$, the property guarantees that $$(3x + y) + (2x - y) = 5 + 0$$, which simplifies to $$5x = 5$$.

Extended Addition Property of Equality

In a corporate accounting scenario, if two separate department budgets are currently equal ($a = b$), the Addition Property of Equality describes the rule for adding the same amount of extra funding ($c$) to both. State the rule of the Addition Property of Equality in your own words and provide the algebraic formula that represents this property using the variables $a, b,$ and $c$.

Maintaining Balance in Financial Accounts

Based on the scenario provided, identify the specific mathematical property being demonstrated and state its formal algebraic definition using the variables a, b, and c.

Equity in Salary Adjustments

Imagine you are developing a 'Best Practices' module for a company's financial training program. Your task is to explain a fundamental principle used to maintain balance in quantitative relationships. Write a brief essay describing the **Addition Property of Equality**. Your response must include:

1. The formal definition of the property in your own words.
2. The algebraic formula representing the property using the variables $a$, $b$, and $c$.
3. An explanation of how this property is used as a functional tool to determine the value of a variable when a value is being subtracted from it.

Ensure your explanation is clear, professional, and follows a logical structure suitable for a corporate training guide.

The Principle of Mathematical Balance in Professional Documentation

After simplifying an equation to the form $$x - 11 = 7$$, the next step is to isolate the variable $$x$$. To undo the subtraction of 11, apply the Addition Property of Equality by adding 11 to both sides of the equation: $$x - 11 + 11 = 7 + 11$$. This operation cancels out the subtraction on the left side, isolating $$x$$, and updates the right side to 18, producing the final solution $$x = 18$$.

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